Stress state analysis of an equal slope shell under uniformly distributed tangential load by different methods

Nowadays there are various calculation methods for solving a wide range of problems in construction, hydrodynamics, thermal conductivity, aerospace research and many other areas of industry. Analytical methods that make up one class for solving problems, and numerical calculation methods that make up another class, including those implemented in computing complexes, are used for the design and construction of various thin-walled structures such as shells. Due to the fact that thin-walled spatial structures in the form of various shells are widely used in many areas of human activity it is useful to understand and know the capabilities of different calculation methods. Research works on the study of the stress-strain state of the torse shell of equal slope with an ellipse at the base are not widely available at the moment. For the first time the derivation of the differential equations of equilibrium of momentless theory of shells to determine the normal force Nu from the action of uniformly distributed load tangentially directed along rectilinear generatrixes to the middle surface of the torse of equal slope with a directrix ellipse is presented in this article. The parameters of the stress state of the studied torse are also obtained by the finite element method and the variational-difference method. The SCAD software based on the finite element method and the program SHELLVRM written on the basis of the variational-difference method are used. The numerical results of the parameters of the stress state of the studied torse are analyzed, and the advantages and disadvantages of the analytical method and two numerical calculation methods are determined.

Approximations of the Sixth Order with the Polynomial and Non-polynomial Splines and Variational-difference Method

This paper discusses the approximations with the local basis of the second level and the sixth order. We call it the approximation of the second level because in addition to the function values in the grid nodes it uses the values of the function, and the first and the second derivatives of the function. Here the polynomial approximations and the non-polynomial approximations of a special form are discussed. The non-polynomial approximation has the properties of polynomial and trigonometric functions. The approximations are twice continuously differentiable. Approximation theorems are given. These approximations use the values of the function at the nodes, the values of the first and the second derivatives of the function at the nodes, and the local basis splines. These basis splines are used for constructing variational-difference schemes for solving boundary value problems for differential equations. Numerical examples are given

Comparative analysis of the results of determining the parameters of the stress-strain state of equal slope shell

Relevance. Thin-walled structures of shells constitute a large class in architecture, in civil and industrial construction, mechanical engineering and instrument making, aircraft, rocket and shipbuilding, etc. Each surface has certain ad-vantages over the other. So the torso surface can be deployed on the plane of all its points without folds and breaks, with the length of the curves and the angles between any curves belonging to the surface, do not change. The investigation of the stressstrain state of the equal slope shell with a director ellipse at the base is presented to date in a small volume. The aim of the work. Obtaining data for comparative analysis of the results of the stress-strain state of equal slope shells by the finite element method and the variational-difference method. Methods. To assess the stressstrain state of the equal slope shell, the SCAD Office computer complex based on the finite element method and the “PLATEVRM” program, written on the basis of the variational-difference method, are used. Results. The numerical results of the stress-strain state of the equal slope shell are obtained and analyzed, the pros and cons of the results of calculations by the finite element method and the variational-difference method are revealed.

STATIC ANALYSIS OF A PLATE ON ELASTIC FOUNDATION WITH THE MODELING IN PC “LIRA” AND WITH THE CALCULATION BY VARIATIONAL-DIFFERENCE METHOD

Static analysis of a plate on elastic foundation is made in two ways on the example of a reinforced concrete road plate 2PP30.18-30 series B3.503.1-1 intended for temporary roads. The plate are calculated on the PC «LIRA». Also the plate are calculated by the variational-difference method like a construction on linearly elastic inhomogeneous foundation (in the form of criss-crossed reinforced concrete tapes). When setting up a contact problem, hypotheses and assumptions of the theory of elasticity and structural mechanics are used. The hypotheses of the theory of bending are valid for a plate. When the deformation energy functional is compiled, the work of the forces of the intrinsic weight of the elastic foundation is not taken into account. In the contact zone, the equality of the displacements of the foundation to the deflections of the plate is valid. When calculating a layered elastic medium is replaced by a rectangular design area, which is approximated by a rectangular grid of finite dimensions. For unknowns are accepted vertical and horizontal movements of the nodes of the calculation area and reactive pressures (contact stresses) in the contact zone of the plate with the base. Based on the calculated values of nodal displacements the intensity of deformations and the intensity of stresses are identifying. When calculating in a PC «LIRA», 6 loading options are considered (central and edge). When calculating the variational-difference method, only central loading is considered. There are given the results of numerical studies in comparison to the two variants of calculations in this article.

FEATURES OF THE NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS WITH THE USE OF PYTHON ALGORITHMIC PROGRAMMING LANGUAGE

The solution of the majority of the practical tasks arising at calculation of constructions demands a numerical solution of boundary value problems. In article on examples of a solution of specific boundary value problems possibilities of a modern object-oriented programming language of Python are described. This lan-guage possesses convenient syntax and opportunities of flexible use of the existing libraries of the numerical methods allowing the user to prepare a program code and to start studying of a task in the shortest terms. Rich opportunities of graphic library Matplotlib are allowed to have graphic display of results of calculation. The Numpy library used in article is standard library for carrying out any matrix and vector calculations in the Python language, its application turns the Python language on Wednesday for programming of numerical calculations, similar to such widely known software products as Matlab and GNU Octave. In article the method of finite differences has solved a task for Laplace's equation, the describing field distribution of temperature on rectangular area; problem of search of a minimum of functionality of Dirikhle by a variational-difference method; a dy-namic problem of vibration of weight on a visco-elastic element at vibrations of the basis and a three-dimensional task of the theory of elasticity by a variational-difference method.